The Forbidden Forest is forbidden, but it doesn't seem to be such a big deal, and its danger seems largely overhyped.
In Philosopher's Stone Dumbledore makes the following announcement:
"First years should note that the forest on the grounds is forbidden to all pupils. And a few of our older students would do well to remember that as well."
Dumbledore's twinkling eyes flashed in the direction of the Weasley twins.
Notice the "twinkling eyes". I would say this is indicative of how the rule is viewed. The forest is officially forbidden, but no one really cares. It's like a game where the mischievous students keep on going in and the staff keeps on telling them off, but both sides know that nothing will change. In fact, one might even say that Dumbledore enjoys it. Note that Dumbledore's educational philosophy is basically to let students break rules and do whatever they want in order to figure things out for themselves.
At the start of term in Order of the Phoenix Dumbledore similarly indicates that some of the students don't care about the rule (and that he probably doesn't care that they break it):
"First years ought to know that the forest in the grounds is out of bounds to students — and a few of our older students ought to know by now too." (Harry, Ron, and Hermione exchanged smirks.)
Hagrid, too, seems to think of it as some sort of game. As he tells Ron the first time they meet:
"Another Weasley, eh?" said Hagrid. "I spent half me life chasin' yer twin brothers away from the forest.
It is also noteworthy to contrast how Dumbledore announces that the forest is forbidden with how he announces that the third floor corridor is forbidden. Regarding the former, as mentioned, he simply stated that it is forbidden — and even this was with twinkling eyes. Regarding the latter, though, he is much more serious:
"And finally, I must tell you that this year, the third floor corridor on the right-hand side is out of bounds to everyone who does not wish to die a very painful death."
Additionally, the door to the third floor corridor is actually locked (albeit a lock that can be opened wit a simple Alohamora), while there is no barrier preventing anyone from entering the forest.
Hagrid takes a group of first year students into the forest and sends two of them off on their own with just a dog that he himself states is a coward:
Right, now, we're gonna split inter two parties an' follow the trail in diff'rent directions. There's blood all over the place, it must've bin staggerin' around since last night at least."
"I want Fang," said Malfoy quickly, looking at Fang's long teeth.
"All right, but I warn yeh, he's a coward," said Hagrid.
If the forest was truly so dangerous, Hagrid would not be able to send off two first years on their own like that. Indeed, he tells them:
an' if anyone gets in trouble, send up red sparks, an' we'll come an find yeh
Now if the two groups are far apart, it would take Hagrid at least a few minutes to come to the rescue. So clearly he is not worried that there is any real danger (i.e. something that would seriously hurt them before he would have time to get to them).
In fact, when you stop to think about it, we never really find any dangers terrorizing the forest. There's Aragog and his clan, but they are deep in the forest, well off the path, and don't seem to bother you unless you stray into their territory. Apart from them there's just the centaurs, who, until Hagrid gets on their bad side in Order of the Phoenix, seem to be perfectly affable to humans. There are, of course, the anomalies such as Voldemort, Grawp, dragons, etc. but those are not things that are inherently part of the forest; they are just things that are temporarily introduced into the forest.
Indeed, as Hagrid explicitly tells Harry, Malfoy, Neville, and Hermione:
"There's nothin' that lives in the forest that'll hurt yeh if yer with me or Fang," said Hagrid.
Note, that despite all this, Mr. and Mrs. Weasley seem to think the forest is actually a big deal. As Mr. says to Mrs. in the beginning of Prisoner of Azkaban:
"You know what Harry and Ron are like, wandering off by themselves — they've even ended up in the Forbidden Forest!
But then again, that Mr. and Mrs. Weasley seem to think that the forest is the primary danger at Hogwarts is somewhat laughable. #CluelessParents